Question

Design of a Tubular Reactor to Precipitate a Protein A protein at a concentration of 2.0 g/liter is to be precipitated in a tubular reactor at 20° C and a rate of 1.0 kg/h. It is desired that the protein precipitate particles leaving the reactor have a diameter of 10 µm. The properties of the protein are as follows: molecular weight of 480,000, diffusion coefficient of 3.5 × 10?7 cm2 /s at 20? C, and precipitate particle density of 1.29 g/cm3 . Design a tubular reactor (i.e., specify diameter and length) to carry out the precipitation in turbulent flow at a Reynolds number of 10,000. It can be assumed that the particle collision effectiveness factor (?) is 0.05 for particle growth governed by fluid motion

Short answer

First, obtain the volume flow rate as 0.5 L/hr by dividing the flow rate by the concentration. Next, calculate the diffusional velocity as \(5.686 × 10^-8 m/s\) using the given diameter and density, and diffusion coefficient. Then, using the diffusional velocity divided by the collision efficiency factor, find the reactor diameter to be \(6.733 × 10^-2 m\) or \(6.73 cm\) and the Reynolds number to be \(1.667 × 10^-6 m^2/s\). Finally, the reactor length is calculated as \(5.463 × 10^3 m\) or \(5.463 km\).

Step-by-step solution

Calculate the Volume Flow Rate

The volume flow rate can be determined using the equation \(Q = \frac{m}{C}\), where \(m = 1.0 kg/h\) and \(C = 2.0 g/L\). Convert \(m\) to grams and \(C\) to kg/L before calculating \(Q\) in L/h.

Calculate the Diffusional Velocity

The diffusional velocity can be determined using the equation \(u_d = \frac{d}{\rho t}\), where \(d = 10 µm\), \(\rho = 1.29 g/cm^3\), and \(t = \frac{1}{D}\) with \(D = 3.5 × 10^-7 cm^2/s\). Convert \(d\) to cm and \(\rho\) to kg/m^3 before calculating \(u_d\) in m/s.

Calculate the Reactor Diameter

The reactor diameter can be determined using the equation \(D = \sqrt{ \frac{4Q}{\pi u}}\), where \(Q\) is determined from step 1 and \(u\) is calculated using the equation \(u = \frac{u_d}{\varepsilon}\) where \(\varepsilon = 0.05\). Convert \(Q\) to m^3/s before calculating \(D\) in m.

Calculate the Reynolds Number

Reynolds number can be determined by the equation \(Re = \frac{uD}{v}\). Given that \(Re = 10000\), we can solve for \(v\).

Calculate the Reactor Length

The reactor length can be calculated by the equation \(L = \frac{Q}{uD \pi}\), where \(Q\) was determined from step 1, \(D\) from step 3, and \(u\) from step 3. Convert \(Q\) to m^3/s before calculating \(L\) in m.

Key concepts

Protein Precipitation

In biochemical engineering, protein precipitation is a critical process used for isolating proteins from a solution. It involves bringing proteins out of a solution form and into a solid, particulate form, often to purify them, concentrate them, or prepare them for further processing like drying. During precipitation, various agents like salts or organic solvents are added to the solution to reduce the solubility of proteins, causing them to aggregate and form precipitates.

Understanding the conditions for precipitation is vital for tubular reactor design, as the size of the protein particles and their density affects how they behave within the reactor. Factors such as the concentration of the protein, temperature, flow rate, and the presence of precipitating agents must be carefully controlled to achieve the desired particle size, in our case, 10 µm. The design must ensure that the conditions inside the reactor allow for efficient particle growth and precipitation while maintaining a controlled environment to avoid undesired by-products or reactions.

Reynolds Number

The Reynolds number (Re) is a dimensionless quantity that helps predict flow patterns in different fluid flow situations. In tubular reactors, it is used to determine the regime of the flow—whether it is laminar or turbulent. The value of the Reynolds number is calculated using the formula \(Re = \frac{uD}{v}\), where \(u\) is the fluid velocity, \(D\) is the diameter of the pipe, and \(v\) is the kinematic viscosity of the fluid.

A Reynolds number of 10,000 indicates that the flow regime inside the reactor is turbulent, which is essential for efficient mixing and particle collisions needed for protein precipitation. The turbulent flow enhances the mass transfer and heat transfer, leading to more uniform particle sizes and potentially higher efficiencies in the precipitation process. When designing our tubular reactor, we must ensure that we achieve this Reynolds number to facilitate proper mixing and reactor performance.

Diffusional Velocity

The concept of diffusional velocity is rooted in the molecular transport of species in a fluid. It can be defined as the velocity at which particles or molecules move from regions of higher concentration to regions of lower concentration owing to their random thermal motion. The relationship between the diameter of the particle \(d\), the density of the precipitate \(\rho\), and the diffusion coefficient \(D\) helps in calculating the diffusional velocity \(u_d\) using the formula \(u_d = \frac{d}{\rho t}\), where \(t\) is the time relating to the diffusion coefficient.

Maintaining appropriate diffusional velocities in a tubular reactor is crucial for achieving the target particle size for protein precipitation. For instance, if the diffusional velocity is too high, it might lead to smaller particles that do not settle well, while too low velocities might result in larger, perhaps unwanted, agglomerates. Therefore, the rate at which proteins move needs to be finely balanced to meet the specific particle size requirements of the reactor's design criteria.

Turbulent Flow

In the context of tubular reactors, turbulent flow is a flow regime characterized by chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs at lower Reynolds numbers and is defined by smooth, constant fluid motion in layers that do not mix. Turbulence is beneficial in many industrial processes, including protein precipitation, because it enhances mixing and mass transfer.

Key to designing reactors with turbulent flow is to ensure the Reynolds number is above a critical threshold, which happens to be approximately 4000 for pipes. Since the exercise aims for a Reynolds number of 10,000, turbulent flow will be guaranteed. This type of flow induces mechanical forces that promote the formation of precipitates by encouraging the collision and coalescence of protein molecules and particles.

Controlling turbulent flow within a tubular reactor is not only critical for precipitate quality but also impacts the reactor design. In our scenario, the diameter and the length of the tubular reactor have to be calculated in such a way that they promote turbulence without causing excessive pressure drops or inefficient processing, which underscores the importance of correctly applying Reynolds number in the design process.